[Note: GERMAN versions of the Week 8 tasks are available HERE. I welcome suggestions on how to improve the translations!]
This week we look at the geometry of the gramophone record, in particular the 45 rpm 7" single and the 33⅓ rpm 12" LP, both of which became popular in the 1950s. (They declined in use from the 1980s with the growth in popularity of the tape cassette and then the CD, whose use in turn declined with the growth of the MP3 player and streaming...)
Monday: The prime purpose of this first task is to remind students of, or more probably introduce them to, the properties of the 7" single and the 12" LP.
The mathematical content of this task - circumference of a circle - is fairly straightforward. The challenge is more in deciding what to do than how to do it. However, the task does involve a rather nice subtlety, which students might not see at first: we don’t need to make use of π, even though we are essentially comparing circumferences. The task can be reduced to deciding which is larger, 12 × 33⅓ or 7 × 45.
The task can also be solved qualitatively: 12 is close to double 7, but 45 is less close to double 33⅓, so the rim of the LP travels faster. Similarly, 12×30, say, is larger than 7×50, say.
Tuesday: A routine task involving ratio notation. Oh well! (The context itself is quite interesting....)
Wednesday: This gets more interesting. Green Onions is a fairly standard single in terms of running time (about 3 minutes). It seems the grooves* are only about 0.21mm apart, or just under 5 grooves per mm, which seems quite tightly packed!
(*We speak of grooves in the plural, though strictly speaking there is only a single groove!)
In 1 minute the single spins 45 full turns, in another minute another 45 turns, in the remaining 56 seconds another 42 turns (45 × 56/60), making 132 turns in all. These are packed into a length of 28mm, so the grooves are about 0.21mm apart (28mm ÷ 132).
Thursday: At the start of the music, the needle travels on a circle (almost) of radius 84mm and at the very end on an almost circle of radius 56mm. If we assume the groove is evenly spaced, we can say the average radius over the 2:56 minutes of playing time is 70mm. This is the key idea to solving the problem.
Do students hit up on this key idea of using an average radius for finding the length of 'each' groove? It is possible that some students opt for this as a 'default' strategy (what else can one do?!) while others, who realise that the speed of the groove under the needle varies, may wonder whether it is legitimate.
We know from Wednesday's task that the disc turns 132 times in the course of the music, so the total distance of the groove that the needle travels over is given by 2𝝅 × 70mm × 132 = 58027.2mm or about 58m.
Friday: The length of the (almost) circles that the groove makes clearly get shorter as the music progresses and the needle moves closer to the centre of the disc. But the needle travels over each (almost) circle in the same length of time (1/45 of 1 minute). So it covers less and less of the track over a given period of time as the music progresses. So, over the first half of the music, it will cover a greater distance than over the second half, so more than 29m.
How readily do students see that when the music is halfway through, the position of the needles is halfway between 84mm and 56mm from the centre of the record, rather than halfway along the music groove?
And if the needle’s speed relative to the surface of the record gets progressively slower, how is it that the pitch of the music doesn’t get progressively lower?!
You might want to challenge students to estimate the actual length of track covered by the needle during the first and second halves of the music.
(The distances are in the ratio 77:63 or 11:9.)
Note: A cassette tape behaves quite differently from a gramophone record - the tape travels across the pick-up head at a constant speed, while the speed at which each of the reels holding the tape rotates, changes continuously.
